The Mobile Radio Propagation Channel P2

There is a concept known as e€ective area which is useful when dealing with antennas in the receiving mode.If an antenna is irradiated by an electromagnetic wave, the received power avai

So, given PTand G it is possible to calculate the power density at any point in the far ®eld that lies in the direction of maximum radiation.A knowledge of the radiation pattern is necessary to determine the power density at other points The power gain is unity for an isotropic antenna, i.e one which radiates uniformly

in all directions, and an alternative de®nition of power gain is therefore the ratio of power density, from the speci®ed antenna, at a given distance in the direction of maximum radiation, to the power density at the same point, from an isotropic antenna which radiates the same power.As an example, the power gain of a half-wave dipole is 1.64 (2.15 dB) in a direction normal to the dipole and is the same whether the antenna is used for transmission or reception

There is a concept known as e€ective area which is useful when dealing with antennas in the receiving mode.If an antenna is irradiated by an electromagnetic wave, the received power available at its terminals is the power per unit area carried

by the wave6the e€ective area, i.e P ˆ WA.It can be shown [1, Ch.11] that the e€ective area of an antenna and its power gain are related by

A ˆl2G

2.2 PROPAGATION IN FREE SPACE

Radio propagation is a subject where deterministic analysis can only be applied in a few rather simple cases.The extent to which these cases represent practical conditions is a matter for individual interpretation, but they do give an insight into the basic propagation mechanisms and establish bounds

If a transmitting antenna is located in free space, i.e remote from the Earth or any obstructions, then if it has a gain GT in the direction to a receiving antenna, the power density (i.e power per unit area) at a distance (range) d in the chosen direction is

The available power at the receiving antenna, which has an e€ective area A is therefore

PRˆP4pdTG2T A

ˆP4pdTG2T l24pGR

where GRis the gain of the receiving antenna

Thus, we obtain

PR

PT ˆ GTGR

l 4pd

 2

…2:3†

which is a fundamental relationship known as the free space or Friis equation [2].The well-known relationship between wavelength l, frequency f and velocity of propagation c (c ˆ f l) can be used to write this equation in the alternative form

PR

PTˆ GTGR

c 4pfd

…2:4†

The propagation loss (or path loss) is conveniently expressed as a positive quantity and from eqn.(2.4) we can write

LF…dB† ˆ10 log10…PT=PR†

ˆ 10 log10GT 10 log10GR‡ 20 log10 f ‡ 20 log10d ‡ k …2:5†

where k ˆ 20 log10 4p

3  108

ˆ 147:56

It is often useful to compare path loss with the basic path loss LBbetween isotropic antennas, which is

LB …dB† ˆ 32:44 ‡ 20 log10 fMHz‡ 20 log10dkm …2:6†

If the receiving antenna is connected to a matched receiver, then the available signal power at the receiver input is PR.It is well known that the available noise power is kTB, so the input signal-to-noise ratio is

SNRiˆ PR

kTBˆ

PTGTGR kTB

c 4p fd

If the noise ®gure of the matched receiver is F, then the output signal-to-noise ratio is given by

SNRoˆ SNRi=F

or, more usefully,

…SNRo†dBˆ …SNRi†dB FdB Equation (2.4) shows that free space propagation obeys an inverse square law with range d, so the received power falls by 6 dB when the range is doubled (or reduces by

20 dB per decade).Similarly, the path loss increases with the square of the transmission frequency, so losses also increase by 6 dB if the frequency is doubled High-gain antennas can be used to make up for this loss, and fortunately they are relatively easily designed at frequencies in and above the VHF band.This provides a solution for ®xed (point-to-point) links, but not for VHF and UHF mobile links where omnidirectional coverage is required

Sometimes it is convenient to write an expression for the electric ®eld strength at a known distance from a transmitting antenna rather than the power density.This can

be done by noting that the relationship between ®eld strength and power density is

W ˆEZ2 where Z is the characteristic wave impedance of free space.Its value is 120p (377 O) and so eqn.(2.2) can be written

E2 120pˆ

PTGT 4pd2 giving

E ˆ



30PTGT p

Finally, we note that the maximum useful power that can be delivered to the terminals of a matched receiver is

P ˆE2A

E2 120p

l2GR 4p ˆ

El 2p

 2

GR

2.3 PROPAGATION OVER A REFLECTING SURFACE

The free space propagation equation applies only under very restricted conditions; in practical situations there are almost always obstructions in or near the propagation path or surfaces from which the radio waves can be re¯ected.A very simple case, but one of practical interest, is the propagation between two elevated antennas within line-of-sight of each other, above the surface of the Earth.We will consider two cases, ®rstly propagation over a spherical re¯ecting surface and secondly when the distance between the antennas is small enough for us to neglect curvature and assume the re¯ecting surface to be ¯at.In these cases, illustrated in Figures 2.1 and 2.4 the received signal is a combination of direct and ground-re¯ected waves To determine the resultant, we need to know the re¯ection coecient

2.3.1 The re¯ection coecient of the Earth

The amplitude and phase of the ground-re¯ected wave depends on the re¯ection coecient of the Earth at the point of re¯ection and di€ers for horizontal and vertical polarisation.In practice the Earth is neither a perfect conductor nor a perfect dielectric, so the re¯ection coecient depends on the ground constants, in particular the dielectric constant e and the conductivity s

For a horizontally polarised wave incident on the surface of the Earth (assumed to

be perfectly smooth), the re¯ection coecient is given by [1, Ch.16]:

rhˆsin c



…e=e0 js=oe0† cos2c p

sin c ‡p…e=e0 js=oe0† cos2c where o is the angular frequency of the transmission and e0is the dielectric constant

of free space.Writing eras the relative dielectric constant of the Earth yields

rhˆsin c



…er jx† cos2c p

sin c ‡p…er jx† cos2c …2:9† where

x ˆoes

0ˆ18  10f 9s For vertical polarisation the corresponding expression is

rvˆ…er j x† sin c p…er jx† cos2c

…er jx† sin c ‡p…er jx† cos2c …2:10† The re¯ection coecients rhand rvare complex, so the re¯ected wave will di€er from the incident wave in both magnitude and phase.Examination of eqns (2.9) and (2.10) reveals some quite interesting di€erences For horizontal polarisation the relative phase of the incident and re¯ected waves is nearly 1808 for all angles of incidence.For very small values of c (near-grazing incidence), eqn.(2.9) shows that the re¯ected wave is equal in magnitude and 1808 out of phase with the incident wave for all frequencies and all ground conductivities.In other words, for grazing incidence

rhˆ jrhjejyˆ 1ej pˆ 1 …2:11†

As the angle of incidence is increased then jrhj and y change, but only by relatively small amounts.The change is greatest at higher frequencies and when the ground conductivity is poor

Figure 2.1 Two mutually visible antennas located above a smooth, spherical Earth of e€ective radius re

For vertical polarisation the results are quite di€erent.At grazing incidence there

is no di€erence between horizontal and vertical polarisation and eqn.(2.11) still applies.As c is increased, however, substantial di€erences appear.The magnitude and relative phase of the re¯ected wave decrease rapidly as c increases, and at an angle known as the pseudo-Brewster angle the magnitude becomes a minimum and the phase reaches 7908.At values of c greater than the Brewster angle, jrvj increases again and the phase tends towards zero.The very sharp changes that occur

in these circumstances are illustrated by Figure 2.2, which shows the values of jrvj and y as functions of the angle of incidence c.The pseudo-Brewster angle is about

158 at frequencies of interest for mobile communications (x  er), although at lower frequencies and higher conductivities it becomes smaller, approaching zero if x  er Table 2.1 shows typical values for the ground constants that a€ect the value of r The conductivity of ¯at, good ground is much higher than the conductivity of poorer ground found in mountainous areas, whereas the dielectric constant, typically 15, can be as low as 4 or as high as 30.Over lakes or seas the re¯ection properties are quite di€erent because of the high values of s and er.Equation (2.11) applies for horizontal polarisation, particularly over sea water, but r may be signi®cantly di€erent from 71 for vertical polarisation

Figure 2.2 Magnitude and phase of the plane wave re¯ection coecient for vertical polarisation.Curves drawn for s ˆ 12  10 3, erˆ 15.Approximate results for other frequencies and conductivities can be obtained by calculating the value of x as 18  103s=fMHz

2.3.2 Propagation over a curved re¯ecting surface

The situation of two mutually visible antennas sited on a smooth Earth of e€ective radius re is shown in Figure 2.1 The heights of the antennas above the Earth's surface are hTand hR, and above the tangent plane through the point of re¯ection the heights are h0T and h0R.Simple geometry gives

d2 ˆ ‰re‡ …hT h0

T†Š2 r2 ˆ …hT h0

T†2‡ 2re…hT h0

T† ' 2re…hT h0

T† …2:12† and similarly

d2' 2re…hR h0

Using eqns (2.12) and (2.13) we obtain

h0

Tˆ hT d21

2re and h0Rˆ hR

d2 2

The re¯ecting point, where the two angles marked c are equal, can be determined by noting that, providing d1, d244hT, hR, the angle c (radians) is given by

c ˆhdT0

1 ˆhd0R 2 Hence

h0 T

h0R'dd1

Using the obvious relationship d ˆ d1+d2 together with equations (2.14) and (2.15) allows us to formulate a cubic equation in d1:

2d3

1 3dd2‡ ‰d2 2re…hT‡ hR†Šd1‡ 2rehTd ˆ 0 …2:16† The appropriate root of this equation can be found by standard methods starting from the rough approximation

d1'1 ‡ hd

T=hR

To calculate the ®eld strength at a receiving point, it is normally assumed that the di€erence in path length between the direct wave and the ground-re¯ected wave is negligible in so far as it a€ects attenuation, but it cannot be neglected with regard to the phase di€erence along the two paths.The length of the direct path is

Table 2.1 Typical values of ground constants

Surface Conductivity s (S) Dielectric constant er Poor ground (dry) 11073 4±7

Good ground (wet) 21072 25±30

R1 ˆ d 1 ‡…hT0 d2h0R†2

and the length of the re¯ected path is

R2 ˆ d 1 ‡…hT0 ‡ hd2 0R†2

The di€erence DR ˆ R2 R1 is

DR ˆ d 1 ‡…hT0 ‡ h0

R†2

d2

1 ‡…h0T h0

R†2

d2

and if d  hT0, hR0 this reduces to

The corresponding phase di€erence is

Df ˆ2pl DR ˆ4phld0Th0R …2:18†

If the ®eld strength at the receiving antenna due to the direct wave is Ed, then the total received ®eld E is

E ˆ Ed‰1 ‡ r exp… j Df†Š

where r is the re¯ection coecient of the Earth and r ˆ j rjexp jy.Thus,

E ˆ Edf1 ‡ jrjexp‰ j…Df y†Šg …2:19† This equation can be used to calculate the received ®eld strength at any location, but note that the curvature of the spherical Earth produces a certain amount of divergence of the ground-re¯ected wave as Figure 2.3 shows This e€ect can be taken into account by using, in eqn.(2.19), a value of r which is di€erent from that derived

in Section 2.3.1 for re¯ection from a plane surface The appropriate modi®cation consists of multiplying the value of r for a plane surface by a divergence factor D given by [3]:

D ' 1 ‡r 2d1d2

e…h0T‡ h0R†

…2:20†

The value of D can be of the order of 0.5, so the e€ect of the ground-re¯ected wave is considerably reduced

2.3.3 Propagation over a plane re¯ecting surface

For distances less than a few tens of kilometres, it is often permissible to neglect Earth curvature and assume the surface to be smooth and ¯at as shown in Figure 2.4 If we also assume grazing incidence so that r ˆ 1, then eqn.(2.19) becomes

E ˆ Ed‰1 exp… jDf†Š

ˆ Ed‰1 cos Df ‡ j sin DfŠ

Thus,

jEj ˆ jEdj‰1 ‡ cos2Df 2 cos Df ‡ sin2DfŠ1=2

ˆ 2jEdjsinDf2 and using eqn.(2.18), with hT0 ˆ hTand h0Rˆ hR,

jEj ˆ 2jEdjsin 2phThR

ld

The received power PRis proportional to E2so

PR/ 4jEdj2sin2 2phThR

ld

ˆ 4PT 4pdl

 2

GTGRsin2 2phThR

ld

…2:21†

If d44hT, hRthis becomes

PR

PTˆ GTGR hTdh2R

…2:22† Figure 2.3 Divergence of re¯ected rays from a spherical surface

Figure 2.4 Propagation over a plane earth

Equation (2.22) is known as the plane earth propagation equation It di€ers from the free space relationship (2.3) in two important ways First, since we assumed that d44hT, hR, the angle Df is small and l cancels out of eqn.(2.22), leaving it frequency independent.Secondly, it shows an inverse fourth-power law with range rather than the inverse square law of eqn.(2.3).This means a far more rapid decrease

in received power with range, 12 dB for each doubling of distance in this case Note that eqn.(2.22) only applies at ranges where the assumption d44hT, hR is valid.Close to the transmitter, eqn.(2.21) must be used and this gives alternate maxima and minima in the signal strength as shown in Figure 2.5

In convenient logarithmic form, eqn.(2.22) can be written

LP …dB† ˆ 10 log10…PT=PR†

ˆ 10 log10GT 10 log10GR 20 log10hT 20 log10hR‡ 40 log10d

…2:23† and by comparison with eqn (2.6) we can write a `basic loss' LBbetween isotropic antennas as

LB …dB† ˆ 40 log10d 20 log10hT 20 log10hR …2:24†

2.4 GROUND ROUGHNESS

The previous section presupposed a smooth re¯ecting surface and the analysis was therefore based on the assumption that a specular re¯ection takes place at the point where the transmitted wave is incident on the Earth's surface.When the surface is Figure 2.5 Variation of signal strength with distance in the presence of specular re¯ection

rough the specular re¯ection assumption is no longer realistic since a rough surface presents many facets to the incident wave.A di€use re¯ection therefore occurs and the mechanism is more akin to scattering.In these conditions characterisation by a single complex re¯ection coecient is not appropriate since the random nature of the surface results in an unpredictable situation.Only a small fraction of the incident energy may be scattered in the direction of the receiving antenna, and the `ground-re¯ected' wave may therefore make a negligible contribution to the received signal

In these circumstances it is necessary to de®ne what constitutes a rough surface Clearly a surface that might be considered rough at some frequencies and angles of incidence may approach a smooth surface if these parameters are changed.A measure of roughness is needed to quantify the problem, and the criterion normally used is known as the Rayleigh criterion.The problem is illustrated in Figure 2.6(a) and an idealised rough surface pro®le is shown in Figure 2.6(b)

Consider the two rays A and B in Figure 2.6(b) Ray A is re¯ected from the upper part of the rough surface and ray B from the lower part.Relative to the wavefront

AA0shown, the di€erence in path length of the two rays when they reach the points

C and C0after re¯ection is

Dl ˆ …AB ‡ BC† …A0B0‡ B0C0†

ˆsin cd …1 cos 2c†

Figure 2.6 Re¯ections from a semi-rough surface: (a) practical terrain situation, (b) idealised model